This work analyzes PINNs for advection-diffusion equations using NTK theory.
problem Understanding and resolving the training difficulties of PINNs for advection-diffusion equations.
method Neural Tangent Kernel (NTK) analysis of PINNs for the linear advection-diffusion equation (LAD).
result PINNs struggle due to spectral bias and convergence rate disparity, especially in advection-dominated and diffusion-dominated regimes.
Study methods to recover unknown processes in PDEs from data.
problem Identifying unknown processes in time-dependent PDEs using observational data.
method Theoretical analysis and numerical approaches including Galerkin and collocation algorithms.
result The Galerkin algorithm is more suitable for practical situations with noisy data.
New method uses PINNs to solve complex PDEs with sparse measurements.
problem Joint estimation of source and parameters in advection-diffusion equations with limited data.
method Weighted adaptive approach based on neural tangent kernel of PINNs.
result Successful estimation of source function, velocity, and diffusion parameters.
The identification of sources of advection-diffusion transport is based usually on solving complex ill-posed inverse models against the available state- variable data records. However, if there are several sources with different locations and strengths, the data records represent mixtures rather than the separate influ…
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
Paper tackles estimating initial conditions of spatio-temporal processes from sparse data.
problem Estimating initial conditions of spatio-temporal advection-diffusion processes from sparse data.
method Regularized convex optimization problem with Alternating Direction Method of Multipliers.
result Efficient solutions for non-uniform and shifted uniform sampling schemes.
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.
A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parame…
We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network approximates the data and constructs the function library, b…
New method converts video of dye plumes into PDEs for better understanding.
problem Inferring continuum models from uncalibrated video data.
method Develops a pipeline to convert grayscale recordings into scalar fields, isolates drift, and identifies transport laws.
result Selected reduced model outperforms advection-diffusion baselines and retains structural interpretability.
Modeling wildfire aerosols using satellite data to predict solar radiation reduction.
problem Accurately estimate and predict AOD propagation from wildfires using multi-source satellite data.
method Physics-informed statistical modeling integrating multi-source satellite data with an advection-diffusion equation.
result The proposed approach accurately predicts AOD propagation and demonstrates model interpretability.
We discuss in this note applications of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) to numerical solutions of partial differential equations arising from stochastic models in quantitative finance. In particular, we develop a framework for solving Black-Scholes-type equations by first t…
Improved DeepONet variants using Transformer cross-conditioning enhance PDE solution efficiency.
problem Solving partial differential equations efficiently and accurately.
method Transformer-inspired DeepONet variants with bidirectional cross-conditioning.
result Improved efficiency and accuracy compared to modified DeepONet, with variant effectiveness tied to PDE characteristics.
We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…
In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form …
Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a…
Clarifies relation for solving control-affine Schrödinger bridge problems.
problem Solving control-affine Schrödinger bridge problems via Hopf-Cole transform.
method Applies Hopf-Cole transform to conditions of optimality, resulting in nonlinear PDEs.
result Generic control-affine Schrödinger bridge requires further algorithmic development.
This paper introduces a linear state-space model with time-varying dynamics. The time dependency is obtained by forming the state dynamics matrix as a time-varying linear combination of a set of matrices. The time dependency of the weights in the linear combination is modelled by another linear Gaussian dynamical model…
In this paper, we consider the use of structure learning methods for probabilistic graphical models to identify statistical dependencies in high-dimensional physical processes. Such processes are often synthetically characterized using PDEs (partial differential equations) and are observed in a variety of natural pheno…
VB-DeepONet uses Bayesian inference to improve DeepONet's predictions and uncertainty quantification.
problem Overfitting and lack of uncertainty quantification in DeepONet.
method Variational Bayes approach to approximate posterior distribution, reducing computational cost.
result VB-DeepONet alleviates DeepONet's limitations and provides uncertainty quantification.
Unified ML approach for SDEs in bounded domains.
problem Challenges in simulating SDEs with particle exit phenomena.
method Hybrid approach combining diffusion model and exit prediction network.
result Accurate modeling of interior dynamics and boundary interactions.
Robust PDE method for path-dependent Asian-style options using MPDATA.
problem Valuation of path-dependent Asian-style options.
method Non-oscillatory forward-in-time second-order MPDATA finite-difference scheme for solving 2D PDEs.
result MPDATA scheme improves solution over first-order upwind step, highlighting its importance.
New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.
problem Unique recovery of transport maps and vector fields from finite measure-valued data.
method Use of Whitney and Takens embedding theorems to establish conditions for unique identification.
result New metric for comparing diffeomorphisms and analogous results in infinitesimal settings.
M-CaStLe discovers causal structures in multivariate space-time data.
problem Challenges in causal graph discovery for high-dimensional gridded data.
method Generalizes CaStLe to multivariate analyses, using local embeddings and pooling spatial replicates.
result More accurately recovers multivariate causal structure and identifies physical dynamics.
SVGP KAN integrates uncertainty quantification into Kolmogorov-Arnold networks.
problem Uncertainty quantification in scientific machine learning models.
method Sparse variational Gaussian process inference with Kolmogorov-Arnold topology.
result Demonstrated ability to distinguish aleatoric and epistemic uncertainty in various scientific applications.
Enhanced DeepONet framework with uncertainty quantification for complex operators.
problem Learning complex operators with uncertainty quantification.
method Generalised variational inference (GVI) using Rényi's α-divergence.
result Superior predictive accuracy and uncertainty quantification.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
Probabilistic grammars improve equation discovery from data.
problem Discovering scientific laws from data using equations.
method Proposed probabilistic context-free grammars to encode soft constraints and a Monte-Carlo algorithm.
result Probabilistic grammars lead to more efficient equation discovery.
In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…